31 Jul Some Favourite Mathematical Constants

Mathematical constant are really exciting and wonderful in the world of numbers. All numbers are not created equal; that certain constants appear at all and then echo throughout mathematics, in seemingly independent ways, is a source of fascination. Just as physical constants provide “boundary conditions” for the physical universe, mathematical constants somehow characterize the structure constants, the Archimedes’ constant (pi) was once regarded as the king, and in modern times (after 1980), the Feigenbaum Universal constant is regarded as the ‘Super King’ in this field on the basis of its increasing importance and tremendous uses in mathematical sciences, physics, chemistry, biosciences, economics, engineering etc.

The constants listed below are rather arbitrarily organized by topicwise. The concrete elaboration of how these constants are derived needs a long discussion, and hence detailed discussion is intentionally omitted. Interested readers are requested to contact the author for details of any constant. However, how the Feigenbaum Universal constant can be obtained is very briefly highlighted below with the help of a simple example.

Let where a is a constant. The interval is mapped into itself by for each value of This family of functions, parameterized by a, is known as the family of logistic maps. What are the 1-cycles (i.e. fixed points) of f? Solving x=f(x) we obtain

x=0 (which attracts for a<1 and repels for a>1),

and (which attracts for 1<a<3 and repels for a>3).

What are the 2-cycles of f? That is, what are the fixed points of the iterate which are not fixed points of f? Solving we obtain the 2-cycle.

(which attracts for and repels for 1+sqrt{6}" />).

For 1+sqrt{6}=3.4495dots" /> an attracting 4-cycle emerges. We can obtain the 4-cycle by numerically solving and It can be shown that 4-cycle attracts for 3.4495….<a<3.5441…., and repels for a>3.5441…..

For a>3.5441…., an attracting 8-cycle emerges. We can obtain the 8-cycle by numerically solving and It can be shown that 8-cycle attracts for 3.5441….<a<3.5644…., and repels for a>3.5644…..

For how long does the sequence of period doubling bifurcations continue? It’s interesting that this behavior stops for short of 4. Setting and so forth denote the cascade of bifurcations, it can be proved that

This point marks the separation between the “periodic regime” and the “chaotic regime” for this family of quadratic functions. The sequence behaves in a universal manner such that the ratio tends to a universal constant

The elementary particle theorist, Mitchell J. Feigenbaum working in the University of Princeton, U.S.A, has explain in details the creation of this constant in his two marvelous papers.

Favourite Mathematical constants so far we know are listed bellow with their approximate numerical values:

Well-known constants

Numerical Value

1

Zero

2

One

1

3

Imaginary unit

4

Pythagoras’ constant

5

Golden mean

6

Natural logarithmic base

C=2.7182818285….

7

Archimedes’ constant

8

Euler-Mascheroni constant

9

Ape’’ry’s constant

10

Catalan’s constant

G=0.915965594….

11

Khintchine’s constant

K=2.68545200….

12

Feigenbaum constant

13

Madelung’s constant

14

Chaiten’s constant

Not available

Constants associate with Number Theory

15

Hardy-Littlewood constant

16

Hadamard-de la Valle’e Poussin constant

17

Landau-Ramanujan constant

K=0.764223653….

18

Brun’s constant

B=1.90211605778….

19

Artin’s constant

20

Linnik’s constant

Not available

21

Hafner-Sarnak-MeCurley constant

22

Gauss-Kuzmin-Wirsing constant

23

Stolarsky-Harborth constant

24

Porter’s constant

C=1.4670780794….

25

Glaisher-Kinkelin constant

A=1.28242713….

26

Franse’n-Robinson constant

2.8077702420….

27

Allodi-Grnstead constant

0.809394020534….

28

Niven’s constant constant

C=1.705211….

29

Backhouse’s constant

1.456074485826….

30

Mill’s constant

C=1.3064….

31

Stieltjes constant

32

Liouville-Roth constant

0.0110001000….

33

Diophantine approximation constant

34

Erdos reciprocal sum constant

3.0089

35

Abundant number density constant

0.2441<A<0.2909

36

Self-number density constant

37

Cameron’s sum-free set constant

0.21759<c<0.21862

38

Euler totient function asymptotic constant

A=1.9435964368….B=-0.0595536246….

39

Nielson-Ramanujan constant

Not available

40

Triple-free set constant

0.6135752692….

41

De-Bruijn-Newman constant

Not yet available

42

Freiman’s constant

Not yet available

43

Cahen’s constant

Not yet available

Constants associate with Analytic Inequalities

44

Shapiro’s cycle sum constant

0.4945668….

45

Carlson-Levin constant

46

Londau-Kolmogorov constant

47

Hilbert’s constant

Not available

48

Copson-de-Bruijn constant

C=1.1064957714….

49

Wirtinger-Sobolev isoperimetric constant

Not available

50

Whitney-Mikhlin extension constant

2.05003

Constants associate with the Approximation of Functions

51

Wilbraham Gibbs constant

G=1.851937052….

52

Lebesgue constant

C=0.9894312738….

53

Favard constant

Not available

54

Bernstein’s constant

55

The “one-ninth” constant

0.1076539192….

56

Laplace limit constant

Constants associate with Enumerating Discrete structures

57

Abelian group enumeration constant

A=2.2948566…. , B=1.3297682….

58

R’enyi’s parking constant

0.7475979203….

59

Golomb Dickman constant

60

Lengyel’s constant

61

Otter’s tree enumeration constant

62

Polya’s random walk constant

63

Self-avoiding-walk connective constant

2.6381585….

64

Feller’s coin tossing constant

65

Har square entropy constant

k=1.503048082….

66

Binary search tree constant

Precise numerical value not available

67

Digital search tree constant

c=0.3720486812….

68

Quardtree constant

C=4.3110704070….

69

Extreme value constant

70

Pattern-free word constant

71

Takeuchi-Prellberg constant

c=2.239433104….

72

Random percolation constant

73

Lenz-Ising constant

74

2D Monomer-dimer constant

1.338515152….

75

3D Dimer constant

76

Lieb’s square ice constant

Not yet available

Constants associate with Functional Iteration

77

Gauss’s lemniscate constant

0.83462684167….

78

Grossman’s constant

Not available

79

Plouffe’s constant

0.4756260767….

80

Lehmer’s constant

0.5926327182….

81

Iterated exponential constant

-0.7666646959….

82

Continued fraction constant

0.76519769…., 0.7098034428….

83

Infinite product constant

2.0742250447….

84

Quadratic recurrence constant

C=1.502836801….

85

Conway’s constant

Constants associate with Complex Analysis

86

Bloch-Landau constant

B=0.4718617….L=0.5432588….

87

Masser-Gramain constant

C=0.6462454398….

88

John constant

4.810477….

Constants associate with Geometry

89

Geometric probability constant

4.2472965….

90

Circular coverage constant

0.8269933431….

91

Universal coverage constant

Not yet available

92

Hermite’s constant

0.7404804897….

93

Tammes’ constant

Not yet available

94

Calabi’s triangle constant

1.5513875245….

95

Graham’s hexagon constant

0.6495190528….

96

Traveling salesman constant

0.521….

97

Moving sofa constant

A=0.09442656084….

98

Beam detection constant

5.1415926536….

99

Heilbronn Triangle constant

H=0.1924500897….

100

Moser’s worm constant

Not yet available

101

Rectilinear crossing constant

0.70449881….

102

Maximum irrodius constant

0.2041241452….

103

Magic geometric constant

0.6675276<m<0.6675284

Almost all the constants seem to be irrationals although rigorous proofs are not available. All the constants have numerous fascinating applications, and thus irrational numbers play very important role in studying modern number theory. Details of some interesting constants and some of their applications will be highlighted in ‘Ganit Bikash’ in near future.